Question: Factor the following expression: $9$ $x^2+$ $17$ $x$ $-2$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(-2)} &=& -18 \\ {a} + {b} &=& & & {17} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-18$ and add them together. Remember, since $-18$ is negative, one of the factors must be negative. The factors that add up to ${17}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${18}$ $ \begin{eqnarray} {ab} &=& ({-1})({18}) &=& -18 \\ {a} + {b} &=& {-1} + {18} &=& 17 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {9}x^2 {-1}x +{18}x {-2} $ Group the terms so that there is a common factor in each group: $ ({9}x^2 {-1}x) + ({18}x {-2}) $ Factor out the common factors: $ x(9x - 1) + 2(9x - 1) $ Notice how $(9x - 1)$ has become a common factor. Factor this out to find the answer. $(9x - 1)(x + 2)$